Found (using google) at least 2 songs featuring "Old Apple Tree"
This is In the shade of... (as mentioned above)
In the Shade of the Old Apple Tree
Words by Harry H. Williams, music by Egbert Van Alstyne, 1905
Popularized by Henry Burr
Performed by Judy Garland and Chorus for the film, Presenting Lily Mars, but cut from the film, 1943 Audio (only) included on laser disc set: Judy Garland: The Golden Years at M-G-M, 1994
Released on CD: Collectors' Gems from the M-G-M Films, 1996
The other was on
http://www.geocities.com/scattershotbyrb/fantastical/the_old_apple_tree.html, the person stated:
" "The Old Apple Tree" is an American folk song. I really don't know how old. Growing up, I often heard the tune used as background music to cartoons made back in the 1930s and 1940s. My grandmother introduced me to the lyrics, which of course went right over my head".
First verse is:
There's an old apple tree in the orchard,
Lives in my memory,
It reminds me of my Pappy,
He was handsome, young and happy
When he planted that old apple tree.